Topological Reorganization and Coordination-Controlled Crossover in Synchronization Onset on Regular Lattices

Abstract: The transition to global synchronization in coupled dynamical systems is governed by the interplay between coupling strength and structural topology. Although abrupt, first-order-like synchronization transitions have been extensively reported in heterogeneous networks, it is unclear whether comparable accelerated onset behavior can emerge purely from coordination geometry in spatially homogeneous, regular lattices. In this study, we investigate large-scale ($N=10^5$) stochastic Stuart-Landau oscillator networks defined on regular lattices with controlled coordination number. Using topological data analysis (TDA), simplicial-complex characterization, and optimal-transport-based geometric diagnostics, we identify a coordination-controlled crossover in synchronization onset dynamics at approximately $z_{c} \approx 7$ within the class of regular lattices considered. Low-coordination lattices ($z < z_{c}$) exhibit persistent $H_2$ topological features in the dynamical amplitude field that correlate with delayed coherence and surface-limited propagation. In contrast, higher-coordination lattices ($z > z_{c}$) display rapid fragmentation of these features, reduced interface roughness, and predominantly positive Ricci curvature. This is consistent with enhanced path redundancy and improved transport efficiency. In this regime, the global order parameter exhibits accelerated exponential-like growth during the onset stage. Throughout this work, abrupt synchronization refers specifically to this exponential onset behavior rather than to thermodynamic first-order hysteresis. Our results demonstrate that increasing coordination density induces a qualitative reorganization of higher-order topological structure that strongly correlates with synchronization efficiency in regular lattice systems.